2010/10/28 by Andersen, Jørgen E., Penner, Robert C., Reidys, Christian M. +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1010.5857
Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus g≥ 0, and we consider the natural generating function \bf Cg[2](z)=∑n≥ 0 \bf c[2]g(n)zn for the number \bf c[2]g(n) of distinct such chord diagrams of fixed genus g≥ 0 with a given number n≥ 0 of chords. We prove here the surprising fact that \bf C[2]g(z)=z2g+1 Rg[2](z)/(1-4z)3g+2 is a rational function, for g≥ 0, where the polynomial R[2]g(z) with degree at most g has integer coefficients and satisfies Rg[2](1\over 4)≠ 0. Earlier work had already determined that the analogous generating function \bf Cg(z)=z2gRg(z)/(1-4z)^3g-1\over 2 for chords attached to a single interval is algebraic, for g≥ 1, where the polynomial Rg(z) with degree at most g-1 has integer coefficients and satisfies Rg(1/4)≠ 0 in analogy to the generating function \bf C0(z) for the Catalan numbers. The new results here on \bf Cg[2](z) rely on this earlier work, and indeed, we find that Rg[2](z)=Rg+1(z) -z∑g1=1g Rg1(z) Rg+1-g1(z), for g≥ 1.